2022-06-07

Plan



The characters:

  • 3-dimensional manifolds and knots
  • 4-dimensional manifolds and knotted surfaces

Why?

How?

  • Measuring knottedness of surfaces via the length of a regular homotopy

Knots in the 3-sphere

Every classical knot \(k \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\) is homotopic to the unknot.



Generically, the homotopy \(H \colon \mathbb{S}^{1} \times [0, 1] \rightarrow \mathbb{S}^{3}\) is a sequence of isotopies and crossing changes.

Knots in other 3-manifolds

4-manifolds and knotted surfaces

Knotted 2-spheres in 4-space: ‘Movies of Links’

Knotted 2-spheres in 4-space: ‘Movies of Links’

Why knotted surfaces?

Knowing how the (immersed) surfaces in a 4-manifold \(M^{4}\) interact can tell us a lot about the topology of \(M\)

  • \(\leadsto\) intersection form \(\pi_{2}(M) \otimes \pi_2(M) \rightarrow \mathbb{Z}\) of \(M\)


Why knotted surfaces?

Given non-closed 4-manifold \(X^{4}\) with 3-manifold boundary \(\partial X = Y\); classical knot \(k \colon \mathbb{S}^{3} \hookrightarrow Y^{3}\),

  • we can investigate surfaces \(\Sigma^{2} \subset X^{4}\) with boundary \(\partial \Sigma = k\)
  • \(\leadsto\) 4-dimensional invariants of \(k\)

Why knotted surfaces?

Related to higher dimensions:

In a 5-dimensional cobordism, the attaching spheres of the 3-handles are 2-knots in 4-manifolds.

Question: How could we define an unknotting number for knotted spheres?

Question: How could we define an unknotting number for knotted spheres?

Question: How could we define an unknotting number for knotted spheres?

(Smale 1958): For every smoothly knotted 2-sphere \(K \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\), there is a regular homotopy starting at the embedding \(K\) and ending at the unknot.

Regular homotopies: Movies of movies

This “movie of movies” is a regular homotopy from a non-trivially knotted 2-sphere (left) to the unknotted sphere in \(\mathbb{S}^{4}\) (right)

Define the Casson-Whitney number \[ \mathop{\mathrm{u_{\mathrm{CW}}}}(K) \text{ of } K \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4} \] as the minimal number of finger moves in a regular homotopy \(K \leadsto\) unknot.


\(\pi K = \pi_{1}(\mathbb{S}^{4} - K) \twoheadrightarrow H_{4}(\mathbb{S}^{4} - K) \cong \mathbb{Z}\cong \langle t \rangle\)

Study the homology of the associated infinite cyclic cover:

\[ \langle t \rangle \curvearrowright \widetilde{(\mathbb{S}^{4} - K)}_{\mathbb{Z}} \rightarrow \mathbb{S}^{4} - K \]

The Alexander module is the following \(\mathbb{Z}[t, t^{1}]\)-module:

\[ \begin{align} H_{1}(\widetilde{(\mathbb{S}^{4} - K)}_{\mathbb{Z}}) & \cong [\pi K, \pi K] \left/ [[\pi K, \pi K], [\pi K, \pi K]] \right. \\ & \cong \pi K^{(1)} \left/ \pi K^{(2)} \right. \end{align} \]

Algebraic effect of a finger move

\[ \pi_{1}(\mathbb{S}^{4} - K') \cong { \pi_{1}(\mathbb{S}^{4} - K) } \; \left/ \; { \langle \langle [ {\color{purple} w}^{{\color{purple} {-1}}} {\color{red} a} {\color{purple} w}, {\color{green} b}] \rangle \rangle } \right. \]



Slogan: Finger moves can make a pair of meridians commute.

\[ { \Large \begin{align} \mathop{\mathrm{u_{\mathrm{CW}}}}(K) & \ge \; \substack{\text{minimal number of finger move relations} \\ [w_{i}^{-1} a_{i} w_{i}, a_{i}] \text{ to make } \pi_{1}(\mathbb{S}^{4} - K) \text{ abelian}} \\[1em] & \ge \; \substack{\text{minimal number of relations} \\ w_{i}^{-1} a_{i} w_{i} = a_{i} \text{ to make } \pi_{1}(\mathbb{S}^{4} - K) \text{ abelian}} \\[1em] & \ge \; \substack{\text{minimal size of generating set} \\ \text{of Alexander module of } K} \end{align} } \]

Results from [Joseph-Klug-R.-Schwartz, 2021]

Prop. \(\mathop{\mathrm{u_{\mathrm{CW}}}}\) can grow arbitrarily large:

There are 2-knots \(K_{n} \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\) with \(\mathop{\mathrm{u_{\mathrm{CW}}}}(K_{n}) = n\).

Beyond the Alexander module

\(K =\) 0-twist spin of \(T_{2, p} \mathop \#T_{2, q}\) is a 2-knot

  • with cyclic Alexander module \(\pi K^{(1)} \left/ \pi K^{(2)} \right.\)
  • but we show that \(\mathop{\mathrm{u_{\mathrm{CW}}}}(K) = 2\)

\(q = p + 2\) or \(q = p + 4\) or (\(q = p + 6\) and \(\gcd(p, p+6) = 1\))


Thm. For \(K_{1}, K_{2}\) knotted surfaces in \(\mathbb{S}^{4}\) with determinant \(\neq 1\), the Casson-Whitney number of the connected sum is \(\mathop{\mathrm{u_{\mathrm{CW}}}}(K_{1} \mathop \#K_{2}) \ge 2\).

Rim surgery: surface \(F^{2} \subset X^{4}\), pattern knots \({\color{blue} J_{\color{blue} 1}}, {\color{blue} J_{\color{blue} 2}} \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\)


(R., 2021): \(\mathop{\mathrm{length_{\textrm{CW}}}}(F_{\tau^{m}}({\color{green} \alpha}, {\color{blue} J_{\color{blue} 1}}), F_{\tau^{m}}({\color{green} \alpha}, {\color{blue} J_{\color{blue} 2}})) \le \mathop{\mathrm{dist_{\textrm{Gord}}}}({\color{blue} J_{\color{blue} 1}}, {\color{blue} J_{\color{blue} 2}})\)

Thm. The Casson-Whitney number \(\mathop{\mathrm{u_{\mathrm{CW}}}}(\tau^{n} k)\) of every twist spin of \(k \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\) is a lower bound for the classical unknotting number of the original knot \(k\).

Special case of (Scharlemann 1985): \(k_{1}, k_{2}\) classical knots with nontrivial determinant, then the classical unknotting number \(u(k_{1} \mathop \#k_{2}) \ge 2\).

Length to ribbon knots

(R., 2021): \(\exists\) a family of 2-knots \(K_{N} \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\) such that the Casson-Whitney length to any ribbon 2-knot \(\rightarrow \infty\).


  • Can take \(K_{N} = \mathop \#^{N} \tau^{0}(k)\)
  • 2-bridge knot \(k \colon \mathbb{S}^{1} \hookrightarrow \mathbb{S}^{3}\) with \(\Sigma_{2}(k) \cong L(p, q)\), \(p\) prime

Suciu’s infinite family of ribbon 2-knots \(R_{l} \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\)

For every \(k \in \mathbb{N}\), \(\pi_{1}(\mathbb{S}^{4} - R_{k})\) is the trefoil knot group

Suciu’s associated \(\mathbb{RP}^{2}\)-knots


(Kanenobu and Sumi 2020): \(R_{k} \mathop \#\mathbb{RP}^{2} \neq R_{l} \mathop \#\mathbb{RP}^{2}\) for \(k \neq l \in \mathbb{N}\)


(R., 2021): For all \(l \ge 1\) we have \(\mathop{\mathrm{length_{\textrm{CW}}}}(R_{l} \mathop \#\mathbb{RP}^{2}, \mathbb{RP}^{2}) = 1\).

Thank you!

References

Freedman, Michael, Robert Gompf, Scott Morrison, and Kevin Walker. 2010. “Man and Machine Thinking about the Smooth 4-Dimensional Poincaré Conjecture.” Quantum Topol. 1 (2): 171–208. https://doi.org/10.4171/QT/5.

Joseph, Jason M., Michael R. Klug, Benjamin M. Ruppik, and Hannah R. Schwartz. 2021. “Unknotting Numbers of 2-Spheres in the 4-Sphere.” J. Topol. 14 (4): 1321–50. https://doi.org/10.1112/topo.12209.

Kanenobu, Taizo, and Toshio Sumi. 2020. “Suciu’s Ribbon 2-Knots with Isomorphic Group.” J. Knot Theory Ramifications 29 (7): 2050053, 9. https://doi.org/10.1142/S0218216520500534.

Klug, Michael, and Benjamin Ruppik. 2021. “Deep and Shallow Slice Knots in 4-Manifolds.” Proc. Amer. Math. Soc. Ser. B 8: 204–18. https://doi.org/10.1090/bproc/89.

Manolescu, Ciprian, Marco Marengon, and Lisa Piccirillo. 2020. “Relative Genus Bounds in Indefinite Four-Manifolds.” https://arxiv.org/abs/2012.12270.

Scharlemann, Martin G. 1985. “Unknotting Number One Knots Are Prime.” Invent. Math. 82 (1): 37–55. https://doi.org/10.1007/BF01394778.

Smale, Stephen. 1958. “A Classification of Immersions of the Two-Sphere.” Trans. Amer. Math. Soc. 90: 281–90. https://doi.org/10.2307/1993205.

Appendix

Regular homotopy for Suciu’s \(\mathbb{RP}^{2}\)-knots

Additional results on the Casson-Whitney number

  • The fusion number is an upper bound on the Casson-Whitney number of ribbon 2-knots.
  • The Casson-Whitney number and the stabilization number are different: \(\exists K \colon \mathbb{S}^{2} \hookrightarrow \mathbb{S}^{4}\) with \(\mathop{\mathrm{u_{\textrm{st}}}}(K) \neq \mathop{\mathrm{u_{\mathrm{CW}}}}(K)\)
  • \(\mathop{\mathrm{u_{\textrm{st}}}}(K) \le \mathop{\mathrm{u_{\mathrm{CW}}}}(K) + 1\)
  • \(\mathop{\mathrm{u_{\mathrm{CW}}}}(K) = 1\) implies \(\mathop{\mathrm{u_{\textrm{st}}}}(K) = 1\)

Deep and shallow slice knots

Comparing topological and smooth sliceness

I found a 4-manifold \(X^{4}\) and non-local knots in \(\partial X\) which are

  • topologically shallow slice,
  • but smoothly deep slice.

Why deep sliceness?

(Freedman et al. 2010)’s attempt at proving a proposed smooth homotopy 4-ball \(\mathcal{B}\) (= smooth contractible compact 4-manifold with \(\mathbb{S}^{3}\) boundary) is exotic:

  • Find knot \(K \subset \mathbb{S}^{3} = \partial \mathcal{B}\) that bounds a smooth slice disk in \(\mathcal{B}\),
  • but \(K\) does not bound any such disk that is contained in a collar \(\mathbb{S}^{3} \times [0, 1]\) of the boundary of \(\mathcal{B}\).

Why deep sliceness?

  • \(X^{4}\) is exotic if there exists a smooth 4-manifold \(X'\) such that
    • \(X\) is homeomorphic to \(X'\),
    • but \(X\) is not diffeomorphic to \(X'\)
  • (Manolescu, Marengon, and Piccirillo 2020):
    • \(X = \#^{3} \mathbb{CP}^{2} \#^{20} \overline{\mathbb{CP}^{2}}\)
    • \(X' = K3 \# \overline{\mathbb{CP}^{2}}\)
    • \(K =\) right handed trefoil
    • 4-manifolds \(X\) and \(X'\) are homeomorphic
    • \(K\) is smoothly \(H\)-slice in \(X\)
    • but \(K\) not smoothly \(H\)-slice in \(X'\).

I found an infinite family of topologically shallow slice and smoothly deep slice knots in the knot trace \(X_{-1}(T_{2, -3})\):

\((p, 1)\)-cable \(J_{p, 1}\) of the knots \[ J = \textrm{Wh}^{(3)}_{+}( \textrm{Wh}_{+}(K^{\ast}, +1), 0 ) \subset \mathbb{S}^{3}_{-1}(T_{2, -3}) = P \]

Top. shallow vs. smooth deep slice for links

(R., 2021): There exists a 2-component link \(J\) in the boundary \(Y = \partial X\) of the 2-handlebody \(X = X_{-1}(T_{2, -3})\) with the following properties:

  • \(J\) is not a local link;
  • \(J\) is link homotopic to the split unlink in the boundary;
  • \(J\) is a boundary link;
  • \(J\) is Brunnian
  • \(J\) bounds smooth strong slice disks in the 2-handlebody \(X\);
  • \(J\) is topologically shallow strong slice in \(Y \times [0, 1]\);
  • \(J\) is not smoothly shallow strong slice in \(Y \times [0,1]\).